LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2012

# MT 3811 – COMPLEX ANALYSIS

Date : 24-04-2012 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

Answer all the questions.

- a) Prove that if using Leibniz’s rule

OR

- b) Let be a non-constant polynomial. Prove that there is a complex number such that. (5)
- c) Let be an analytic function. Prove that for where Hence prove that if f is analytic in an open disk then prove that for where.

OR

- d) State and prove homotopic version of Cauchy’s theorem. (15)

- a) State and prove Morera’s theorem.

OR

- b) Prove that a differentiable function on is convex if and only if is

increasing. (5)

- c) State and prove the Arzela-Ascoli theorem.

OR

- d) State and prove the Riemann mapping theorem.

(15)

- a) Show that in the usual notation.

OR

- b) If and then prove that .

(5)

- c) (i) Let be a compact metric space and let be a sequence of continuous functions from
*X*into such that converges absolutely and uniformly for*x*in*X*. Then prove that the product converges absolutely and uniformly for*x*in*X*. Also prove that there is an integer such that if and only if for some*n*, .

(ii) State and prove Weierstrass factorization theorem. (7+8)

OR

- d) Let , then prove that converges absolutely if and only if converges absolutely.
- e) State and prove Bohr-Mollerup theorem. (7+8)

- a) State and prove Jensen’s formula.

OR

- b) If is a metric space, then prove that is also a metric on (5)
- c) State and prove Rung’s theorem. (15)

OR

- d) State and prove Hadamard’s factorization theorem. (15)

- a) Prove that any two bases of a same module are connected by a unimodular transformation.

OR

- b) Prove that an elliptic function without poles is a constant.

(5)

- c) (i) Prove that the zeros and poles of an elliptic function satisfy .

(ii) Derive Legendre’s relation (7+8)

OR

- d) (i) State and prove the addition theorem for the Weierstrass

(ii) Show that

(8+7)

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